Calculate radius of earth using a pole

[Mk]

Homework Statement

**12. A physicist plants a vertical pole at the waterline on the shore of a calm lake. When she stands next to the pole, its top is at eye level, 175 cm above the waterline. She then rows across the lake and walks along the waterline on the opposite shore until she is so far away from the pole that her entire view of it is blocked by the curvature of the surface of the lake, that is, the entire pole is below the horizon (Figure 1.18). She finds that this happens when her distance from the pole is 9.4 km. From this information, deduce the radius of the Earth.2/3. Relevant equations, solution attempt
My attempt at a diagram. One of the green parts is approximated:
http://img.skitch.com/20091025-k8cm5ie32ifdecsm3fa49y3d3e.jpg

I tried a number of approaches. At first I didn't know whether to think of the 9400 meters in terms of length of a curve, or in terms of a straight line, but then I decided it was negligible considering the solution. I tried working backwards from the arc length. I tried SOHCAHTOA and the Pythagorean Theorem. Not sure quite what to do here. What is wrong about how I am thinking?

Thank you.

 

[Seannation]

Interesting problem.

I think the solution might be found by working out the angle between the woman, the top of the pole and the bottom of the pole. You're given the distance between the woman and the bottom of the pole. The distance between the woman and the top of the pole can be found using Pythagoras' Theorem. The angle between the two distances is related to the curvature of the earth. Dividing 360 degrees by this angle will give you the number of these chords which form the entire circumferance of the earth, and from this you can deduce the radius.

There might be a better way to do this as this method is inaccurate due to the approximations made.

 

[ehild]

I do not understand your diagram. Did you take into account that the pole just can not be seen from the boy's eye level? It is easier so find the solution if you imagine a very high pole and an equally high boy.
Next hint: When the boy walks away from the pole he sees it last time when the light ray traveling from the pole to his eyes just touches the surface of Earth.

ehild

 

[Mk]

I think the solution might be found by working out the angle between the woman, the top of the pole and the bottom of the pole. You're given the distance between the woman and the bottom of the pole. The distance between the woman and the top of the pole can be found using Pythagoras' Theorem. The angle between the two distances is related to the curvature of the earth. Dividing 360 degrees by this angle will give you the number of these chords which form the entire circumferance of the earth, and from this you can deduce the radius.

There might be a better way to do this as this method is inaccurate due to the approximations made.

Nice try, but I don't think that works at all. I did this:

\tan \theta = \frac {\textrm{9400}} {\textrm{1.75}}

\theta \approx 89.989

\frac {360} {\theta} \approx 4.000

So, the circumference of the Earth is 4*9400 m = 37.6 kilometers.

I do not understand your diagram. Did you take into account that the pole just can not be seen from the boy's eye level? It is easier so find the solution if you imagine a very high pole and an equally high boy.
Next hint: When the boy walks away from the pole he sees it last time when the light ray traveling from the pole to his eyes just touches the surface of Earth.

ehild

Okay, I think I solved it!

http://img.skitch.com/20091025-b3pmg299gqtc7s5mkfu5jbtpxx.jpg

I just had to think of it in a new way. Do you think this method is correct? The final answer is nearby to the real value and I think I made no errors. Thank you ehild!

 

[ehild]

The drawing is OK, the method is correct, but you have mixed a bit Pythagoras' Theorem, so the real result is a negative value for the radius of Earth. Moreover, the height of the person is 1.75 m. Try again. (my result is 6311 km.)

ehild

 

[NotaPhysicist]

MK, that final solution is right. And the final diagram is right too. The problem and diagram is in Ohanian Physics, page 20. Pythagoras' theorem does the job.